Sounds like it has one of those flat rectangular lithium polymer batteries that is held in with the sticky adhesive and soldered in. They usually have a 6 digit number which specifies the dimensions. I am hoping there are screws or snaps on the case so it can be opened non-destructably.You could try partial disassembly and maybe read what is printed on the battery for clues. Though, you might have to duct tape it back together if you do find a replacement battery
The→ Remote Microphone + Operations Manual - Starkey says
For Disposal/Recycling Personnel Only
This product contains a Lithium Ion Polymer battery. To remove this battery:
• Power off the Remote Mic +.
• Pry apart the housing halves with a small screwdriver.
• Cut the battery wires from the circuit board to prevent accidental shorting.
• Pry the battery away from the housing using a wide flat blade.